Orbits and Hamilton bonds in a family of plane triangulations with vertices of degree three or six

TL;DR

This paper studies a special family of plane triangulations with vertices of degree three or six, establishing arithmetic relations for their structural parameters, characterizing certain orbits, and proving properties about Hamilton bonds and disjoint paths.

Contribution

It introduces arithmetic equations to compute graph parameters, characterizes one point orbits, and proves new properties of Hamilton bonds and disjoint paths in these triangulations.

Findings

Characterization of one point orbits in the family.

Existence of Hamilton bonds with equitable 2-colorable end-trees for graphs of order 4n+2.

Presence of disjoint induced paths spanning the graph under certain conditions.

Abstract

Let $\cal{P}$ be the family of all 2-connected plane triangulations with vertices of degree three or six. Gr\"{u}nbaum and Motzkin proved (in the dual terms) that every graph $P \in \cal{P}$ is factorable into factors $P_0$, $P_1$, $P_2$ (indexed by elements of the cyclic group $Q = \{0,1,2\}$) such that every factor $P_q$ consists of two induced paths with the same length $M(q)$, and $K(q)-1$ induced cycles with the same length $2M(q)$. For $q \in Q$, we define an integer $S^+(q)$ such that the vector $(K(q), M(q), S^+(q))$ determines the graph $P$ (if $P$ is simple) uniquely up to orientation-preserving isomorphism. We establish arithmetic equations that will allow calculate the vector $(K(q+1), M(q+1), S^+(q+1))$ by the vector $(K(q), M(q), S^+(q))$, $q \in Q$. We present some applications of the equations. The set $\{(K(q), M(q), S^+(q)): q \in Q\}$ is called the orbit of $P$. We characterize one point orbits of graphs in $\cal{P}$. We prove that if $P$ is of order $4n +2$, $n \in\mathbb{N}$, than it has a Hamilton bond such that the end-trees of the bond are equitable 2-colorable and have the same order. We prove that if $M(q)$ is odd and $K(q) \geqslant \frac{M(q)}{3}$, then there are two disjoint induced paths of the same order, which vertices together span all of $P$.